Empty Cells and Nonestimable Interaction Effects

Why Empty Cells Matter

A factorial interaction describes how a factor effect varies across levels of another factor. To estimate such a change from observed data, the relevant factor combinations must provide enough information. When a combination has no observations, some contrasts may be impossible to determine from the design, even if software prints a coefficient for them.

The key question is not simply whether a coefficient appears in the model formula. It is whether the contrast of interest has the same value for every parameter vector that produces the same linear predictors at the observed design points. This definition makes estimability a property of the design and the requested quantity, not of a particular choice of factor coding.

In a generalized linear model, the linear predictor is \(\eta=X\beta\). The criterion below therefore applies to linear contrasts of the coefficients and to linear contrasts of cell-level linear predictors. It does not, by itself, say that every nonlinear function of the parameters is estimable.

The Design-Matrix Criterion

The row space of \(X\) is the set of all linear combinations of its rows; the null space of \(X\) is the set of vectors \(v\) such that \(Xv=0\). The following result turns the definition into a practical test.

Proof. Suppose first that \(c\) belongs to the row space of \(X\). Then there is a vector \(a\) such that \(c=X^\mathsf{T}a\). For any two parameter vectors with \(X\beta=X\tilde{\beta}\),

So the function is estimable. Conversely, suppose \(c^\mathsf{T}\beta\) is the same for every pair of parameter vectors with the same fitted linear predictor. For any \(v\) satisfying \(Xv=0\), the vectors \(\beta\) and \(\beta+v\) have identical predictors, since \(X(\beta+v)=X\beta\). Estimability then requires

Thus \(c\) is orthogonal to the null space of \(X\). The fundamental theorem of linear algebra gives that the orthogonal complement of the null space of \(X\) is the row space of \(X\). Hence \(c\) belongs to the row space. This proves both equivalences. \(\square\)

This criterion also explains why full column rank is sufficient but not necessary for a particular contrast to be estimable. If \(X\) has full column rank, its null space contains only zero, so every linear function of \(\beta\) is estimable. If \(X\) is rank deficient, some functions may still be estimable; the test must be applied to the specific contrast.

Empty Cells in a Saturated Cell-Means Model

For a factorial design, a useful starting point is the saturated cell-means model. It assigns a separate linear predictor \(\eta_j\) to every possible factor combination \(j\). With this parameterization, a row of \(X\) for an observation in cell \(j\) has a 1 in column \(j\) and 0s elsewhere. Repeated observations in a cell repeat the same row. A cell with no observations contributes no row.

Proof. Write the vector of all cell-level linear predictors as \(\eta=(\eta_1,\ldots,\eta_J)^\mathsf{T}\). For each observed cell \(j\), the design includes a row equal to the \(j\)th coordinate vector. If cell \(j\) is empty, no row has a nonzero entry in coordinate \(j\). Consequently, the null space of this cell-indicator design consists exactly of vectors that are zero in observed-cell coordinates and can be arbitrary in empty-cell coordinates. By the design-matrix criterion, a contrast \(d^\mathsf{T}\eta\) is estimable exactly when \(d\) is orthogonal to every such vector. That holds precisely when \(d_j=0\) for every empty cell \(j\). \(\square\)

In this saturated model, observed cell predictors can be estimated separately, but an empty cell's predictor cannot be recovered from the observed cells. An interaction contrast that includes an empty-cell predictor with a nonzero weight is therefore not estimable. This conclusion concerns the saturated model; a restricted model, such as one that assumes additivity, may make additional predictions possible.

Finding Which Contrasts Survive

For several factors or many levels, it is usually easiest to work with the observed design matrix rather than to reason from the word “interaction” alone. In a cell-means design, write down one coordinate for each possible combination, then mark which combinations occur. For a general contrast \(d^\mathsf{T}\eta\), check whether its weights on empty cells are zero. With a non-saturated parameterization, use the row-space or null-space criterion for its coefficient vector.

What a Model Restriction Changes

Nonestimability is relative to a specified model. The saturated cell-means model makes no relationship assumption between different cells. A smaller model can impose such a relationship and thereby identify a missing-cell prediction. This is not new information supplied by the empty cell: it is a consequence of the restriction.

Rank, Coding, and Practical Interpretation

The rank of the design matrix counts the number of independent directions in parameter space that the observed design informs. An empty cell can lower that rank in a saturated model. However, a rank deficiency does not automatically make every coefficient or every scientific contrast nonestimable. The correct question is always whether the particular contrast lies in the row space of the observed design.

Changing the reference level or using effect coding changes the coordinates used to express the model, but it does not add observations or new design information. Under an invertible reparameterization, the same fitted linear predictors and the same estimable scientific quantities remain available. A newly printed coefficient is not evidence that an unobserved interaction has become estimable.

In practice, make the observed-cell pattern explicit before interpreting an interaction. Then express the scientific question as a contrast, check its weights on empty cells in a saturated cell-means representation, or use the equivalent design-matrix test. If a contrast fails, distinguish two possibilities: the contrast is unavailable under the chosen model, or a stated restriction has made it available by connecting cells. That distinction matters for both interpretation and reporting.

Common Mistake

• Wrong reasoning: “The factorial formula contains an interaction term, so the interaction is estimated, even though one combination is absent.” A coefficient can be included in a parameterization without being uniquely determined by the observed design.
• Correction: Check estimability against the actual design matrix. If two coefficient vectors give the same observed linear predictors but different values of the proposed interaction contrast, that contrast is not estimable under the model. A restriction such as additivity can yield a missing-cell prediction, but it changes the model assumptions; it does not recover information from the empty cell.